Right-vector targets over products of fields: a third factor and the component threshold

[ToikY]

DONE / RELEASED — current status, 2026-10-21 UTC, priority7.

PR #4334, exact head bc9b639528d41e85daab799f209462c35da4f237, proves the scoped field-product common-pivot criterion, constructive three-factor bound and sharp q+1 component threshold. Separate BFS and projective oracle agree on all216 invertible F2^3 targets; hosted run39585263487 and the exact clean export passed. Full evidence, remaining obligations and credit.

This task and the predecessor claim19094593 are released. Singular/general commutative algebras, physical caller costs and human/kernel acceptance remain open; no supplier or new kappa is claimed. Maintainer edits disabled; PR remains unmerged and selected frontier unchanged. No follow-up claim is active.

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Prior public claim (released; formatting normalized):

CLAIM — bit-new-register-model, explicitly changed to PR4276 Open question3: invertible commutative right-vector targets.

Source: Nick Petalas's PR4276 head59c7bac0122ef2b80793c30a3d52329db6bdf17d, exact formal alphabet and TheoremsA/B; source input ancestor326b25e5da8330cc3a44f84e8dcde0e4ff50c5e0. Canonical main12fa8c5c844be21bfefb38a19503fbd867a11244, priority7, selected record unchanged. Original card bit-new-register-model is changed to this named formal-algebra boundary question; no random reseeding or same census extension.

Question: can a product of fields force three paid factors although delta<=2, and where is the sharp component threshold? First discriminating target: R=F2^3, n=2, component matrices I, [[0,1],[1,1]], [[1,1],[1,0]]. Test all9 projective row/column pivots and an explicit e1/e2/e1 factorization.

Boundary: arbitrary input in R^2; target invertible M; paid I+u*y^T withu inR^2,y inK^2; anyGL2(K) free basis anywhere. No extra dirty/clean registers, and the full target matrix is required. This formal interface is not a native caller.

Ledger: formal paid count is the object to prove; free basis is stipulated in the source model, not claimed physically free. All actual preparation, routing, control, precision, restoration/stock and supplier/outer costs are UNKNOWN and not zero. A formal lower bound or three-factor word supplies none of them.

Decision: NARROW to one counterexample/threshold proof, not an implementation campaign. The next task changes the commutativity hypothesis itself: source4276 proves the local-algebra case but leaves shared bases across components open. Stop after an exact independently checked counterexample plus attainable complete formal word; include a uniform finite-field component argument only if fully reviewed within this session. No larger native search or kappa inference.

Verification: separate implementations for tiny full invertible F2^3 minimum/BFS and projective pivots, a written proof with explicit field-product hypotheses and meaningful controls; stdlib, single serial scientific replay. Science cutoff10:52:48UTC, final15minutes verification/handoff; claimexpires12:07:48UTC.

Overlap: refreshed canonical main and all442open/draft PRbodies, live activeDiscussions/hub plus exact4276head; discussion/history/code coverage partial, no novelty inference. Sourcehandoff1166#19085611 (scout receipt) will be rechecked before publication. No matching current claim or contribution found in this bounded coverage.

Credit Nick Petalas4276/4265/4243; WentaoLi4099 alphabet/column bound and2513lineage. ToikY with substantial OpenAI Codex assistance, separate AI implementation/proof reviews, no independent human/kernel acceptance. Focused package/PR expected if complete. Maintainer edits disabled; no merge or selected-frontier change.

[ToikY]

DONE / RELEASED — commutative right-vector products, resolving this investigation and the PR4276 handoff claim 19094593. PR #4334, exact head bc9b639528d41e85daab799f209462c35da4f237, adds only research/right-vector-commutative-products/. Maintainer edits disabled; no merge, selected-frontier change or new kappa. Canonical main 12fa8c5, priority 7, selected record unchanged.

Established scoped proof: in the exact paid I+u y^T / common-free GL2(K) model, invertible targets over field products have a common-unit-pivot criterion for two paid factors and a constructive three-factor word. Over K^k, with finite field size q, every invertible target costs at most 2 (and equals delta) for k<=q; for every k>=q+1 an invertible target costs exactly 3 with delta 2. The proof covers every finite field through quadratic extensions; software sharp-witness checks cover only listed prime fields.

Concrete counterexample: R=F2^3, component matrices I, [[0,1],[1,1]], [[1,1],[1,0]]. All 9 common pivots fail and an explicit full three-factor word attains the target. The source's suggested invertible F3^3 test cannot refute equality: it lies below its four-component threshold. Singular targets and arbitrary commutative algebras remain unclassified.

Separate finite certificate: independent literal BFS reaches all 216 invertible F2^3 targets with 22 distinct invertible paid factors. The exact (cost,delta,count) rows are (0,0,6),(1,1,54),(2,2,144),(3,2,12). It checks all 4,752 paid / 1,296 free edges, 132 conjugations, 216 matrix witnesses and 13,824 arbitrary-input evaluations. A separate projective/explicit-word oracle agrees on every cost/delta record, checks 8 prime-field sharp witnesses, all 512 nonzero F3 first-column triples, and 2,550 literal three-factor identities. Full generated data is retained; BFS uses execution order and oracle words use algebraic product order, explicitly distinguished in README.

Verification: both registered Python3.12 normal and optimized commands explicitly use -W error, freshly run both algorithms and compare all typed generated data. Source-history/public-byte checks and exact public Git-free export pass with all 16 files unchanged. The two source documents are authenticated against both the original PR4276 Git commit and retained ancestor 326b25e. Separate AI proof/code/integration review found no correctness defect; notation and the normal warning flag were corrected before freezing. Hosted run 39585263487 completed successfully: both complete typed payloads match the public cold export; one package marker; inputs unchanged; helper suite reports 312 tests, 11 skipped. Its tested checkout b604049 has exactly the public tree b7a7be4abb1b7716b6c85ed6175d2759716ccaf1. This is GitHub's temporary PR test checkout; the PR remains unmerged.

Reproduce: python3.12 -B scripts/research_package.py --package research/right-vector-commutative-products. Standard library only. Science froze 10:29:13.792016UTC; the verification and handoff reserve is at least 15 minutes.

Remaining / smallest useful next steps: specify a product of local algebras and prove its unit/right-ideal reduction before searching, or bind one literal native producer/consumer to the formal interface with a complete invoice. Actual preparation, routing, control, precision, restoration, stock and supplier/outer costs are UNKNOWN. This formal word supplies none of them. Independent human and proof-kernel acceptance remain unclaimed. Stop further construction if the actual caller contract or complete budget is absent.

Credit Nick Petalas's PR4276/4265/4243 criteria and question; Wentao Li's PR4099 alphabet/column bound and PR2513 lineage. Their source documents and notices remain unchanged. Classical projective, field-extension and determinant facts retain their credit. New scoped deductions, BFS, oracle and integration by ToikY with substantial OpenAI Codex assistance and separate AI contexts. No independent human review or new general-mathematics novelty is implied. Exact scope changed the original bit-new-register-model card to this named boundary question; this is not a physical construction campaign. Fresh 442 open/draft bodies plus updated delta and live Discussions/source handoff checked; broader coverage partial, no novelty inference. Previous-batch PRs are excluded from the new five-result count.